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Popular Calculus >

integral of x/(5+x)

Solution

\int \frac{x}{5 + x} d x

Solution

5 + x - 5 ln ∣ 5 + x ∣ + C

Solution steps

\int \frac{x}{5 + x} d x

Apply u-substitution

= \int \frac{u - 5}{u} d u

Expand

\frac{u - 5}{u} : 1 - \frac{5}{u}

= \int 1 - \frac{5}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d u - \int \frac{5}{u} d u

\int 1 d u = u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

= u - 5 ln ∣ u ∣

Substitute back

u = 5 + x

= 5 + x - 5 ln ∣ 5 + x ∣

Add a constant to the solution

= 5 + x - 5 ln ∣ 5 + x ∣ + C

Graph

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Frequently Asked Questions (FAQ)

What is the integral of x/(5+x) ?
The integral of x/(5+x) is 5+x-5ln|5+x|+C